use identities to find values of the sine and cosine functions for the angle measure.\nθ, given that cos 2θ…

use identities to find values of the sine and cosine functions for the angle measure.\nθ, given that cos 2θ = 40/41 and 0° < θ < 90°\nsinθ = √82/82\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)\ncosθ = □\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

use identities to find values of the sine and cosine functions for the angle measure.\nθ, given that cos 2θ = 40/41 and 0° < θ < 90°\nsinθ = √82/82\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)\ncosθ = □\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Answer

Explanation:

Step1: Use the double - angle identity for cosine

The double - angle identity for cosine is (\cos2\theta = 1 - 2\sin^{2}\theta=2\cos^{2}\theta - 1). We can use (\cos2\theta=2\cos^{2}\theta - 1) to find (\cos\theta). Given (\cos2\theta=\frac{40}{41}), then (2\cos^{2}\theta-1 = \frac{40}{41}).

Step2: Solve for (\cos^{2}\theta)

Add (1) to both sides of the equation (2\cos^{2}\theta-1=\frac{40}{41}): (2\cos^{2}\theta=\frac{40}{41}+1=\frac{40 + 41}{41}=\frac{81}{41}). Divide both sides by (2): (\cos^{2}\theta=\frac{81}{82}).

Step3: Solve for (\cos\theta)

Since (0^{\circ}<\theta<90^{\circ}), (\cos\theta>0). Take the square root of (\cos^{2}\theta=\frac{81}{82}), so (\cos\theta=\sqrt{\frac{81}{82}}=\frac{9}{\sqrt{82}}=\frac{9\sqrt{82}}{82}).

Answer:

(\cos\theta=\frac{9\sqrt{82}}{82})